Abstract
In this paper, we establish the general solution and the generalized Hyers–Ulam–Rassias stability problem for a cubic Jensen–type functional equation,
in the spirit of D. H. Hyers, S. M. Ulam, Th. M. Rassias and P. Găvruta.
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References
Ulam, S. M.: A collection of Mathematical Problems, Interscience Publ., New York, 1960
Gruber, P. M.: Stability of isometries. Trans. Amer. Math. Soc., 245, 263–277 (1978)
Zhou, D. X.: On a conjecture of Z. Ditzian. J. Approx. Theory, 69, 167–172 (1992)
Hyers, D. H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci., 27, 222–224 (1941)
Rassias, Th. M.: On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc., 72, 297–300 (1978)
Gajda, Z.: On stability of additive mappings. Internat. J. Math. and Math. Sci., 14, 431–434 (1991)
Găvruta, P.: A generalization of the Hyers–Ulam–Rassias Stability of approximately additive mappings. J. Math. Anal. Appl., 184, 431–436 (1994)
Hyers, D. H., Isac, G., Rassias, Th. M.: Stability of Functional Equations in Several Variables, Birkh¨auser, Basel, 1998
Hyers, D. H., Rassias, Th. M.: Approximate homomorphisms. Aequationes Math., 44, 125–153 (1992)
Rassias, Th. M.: On the stability of the quadratic functional equations and its applications, Studia Univ. “Babes Bolyai”, Mathematica, XLIII, 89–124, 1998
Lee, Y. H., Jun, K. W.: A generalization of the Hyers–Ulam–Rassias stability of Jensen’s equation. J. Math. Anal. Appl., 238, 305–315 (1999)
Trif, T.: Hyers–Ulam–Rassias stability of a Jensen type functional equation. J. Math. Anal. Appl., 250, 579–588 (2000)
Trif, T.: On the stability of a functional equation deriving from an inequality of Popoviciu for convex functions. J. Math. Anal. Appl., 272, 604–616 (2002)
Park, C. G.: Universal Jensen’s equations in Banach modules over a C*–algebra and its unitary group. Acta Mathematica Sinica, English Series, 20(6), 1047–1056 (2004)
Aczél, J., Dhombres, J.: Functional Equations in Several Variables, Cambridge Univ. Press, 1989
Czerwik, S.: The stability of the quadratic functional equation, in ‘Stability of Mappings of Hyers–Ulam Type’ (edited by Th. M. Rassias and J. Tabor), Hadronic Press, Florida, 81–91, 1994
Jun, K. W., Lee, Y. H.: On the Hyers–Ulam–Rassias stability of a pexiderized quadratic inequality. Math. Ineq. Appl., 4(1), 93–118 (2001)
Rassias, Th. M.: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl., 251, 264–284 (2000)
Lee, Y. W.: On the stability of a quadratic Jensen type functional equation. J. Math. Anal. Appl., 270, 590–601 (2002)
Trif, T.: Hyers–Ulam–Rassias stability of a quadratic functional equation. Bull. Korean Math. Soc., 40, 253–267 (2003)
Rassias, J. M.: On the stability of the Euler–Lagrange functional equation. Chinese J. Math., 20, 185–190 (1992)
Rassias, J. M.: Solution of the Ulam stability problem for Euler–Lagrange quadratic mappings. J. Math. Anal. Appl., 220, 613–639 (1998)
Jun, K. W., Kim, H. M.: The generalized Hyers–Ulam–Rassias stability of a cubic functional equation. J. Math. Anal. Appl., 274, 867–878 (2002)
Jun, K. W., Kim, H. M.: Ulam stability problem for a mixed type of cubic and additive functional equation. Bull. Belgian Math. Soc., to appear
Jun, K. W., Kim, H. M.: On the Hyers–Ulam–Rassias stability of a general cubic functional equation. Math. Ineq. Appl., 6(1), 289–302 (2003)
Jun, K. W., Kim, H. M., Chang, Ick–Soon: On the Hyers–Ulam stability of an Euler–Lagrange type cubic functional equation. J. Comput. Anal. Appl., 7(1), 21–33 (2005)
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This work is supported by the Korea Research Foundation Grant funded by the Korea Government (MOEHRD) (KRF–2005–070–C00009)
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Jun, KW., Kim, HM. Stability Problem for Jensen–type Functional Equations of Cubic Mappings. Acta Math Sinica 22, 1781–1788 (2006). https://doi.org/10.1007/s10114-005-0736-9
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DOI: https://doi.org/10.1007/s10114-005-0736-9